17.7 What form does the portfolio frontier take when the N risky assets available to investors...

17.7 What form does the portfolio frontier take when the N risky assets available to investors are mutually uncorrelated?

17.8 Now consider Exercise 17.7 again, but assuming that there are only three assets, that initial wealth is e1m, but that short-selling is not allowed.

(a) Show that the investor’s problem can be formulated as a Kuhn–Tucker maximization problem with inequality constraints, involving either

(i) three choice variables and six inequality constraints, or

(ii) two choice variables and four inequality constraints.

(b) Now consider the specific example in which the expectation of the (net) return vector is

(0.04, 0.08, 0.12) and its variance–covariance matrix is

⎡ 0.03 −0.01 −0.01⎤

⎣−0.01 0.03 −0.01⎦

−0.01 −0.01 0.03

Calculate the optimal investment proportions for desired expected returns of 4%, 5.5%, 6%, 8%, 10%, 11.5% and 12%. What are the signs of the Kuhn–Tucker multipliers in each case (strictly positive, zero, or strictly negative)?

(c) Using the envelope theorem, calculate the rate of change of the minimized variance with respect to the desired expected return in the situation where only the expected return constraint and the budget constraint are binding.

(d) Sketch rough graphs of the feasible set both in the b1b2 plane (i.e. using the two-variable, four-constraint approach) and in the plane b1 + b2 + b3 = 1 (i.e. using the three-variable, six-constraint approach) for two possible values of μ, indicating on each graph the

indifference curves of the objective function.

(e) Over what range of desired expected returns is each short-selling constraint binding? Using this information, plot the envelope function for this problem.

(f) How would your answers in part (b) above change if all the covariances were 0.01 instead of −0.01? Explain.

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